To be precise about the topology, we consider the tiles of a polyiamond to be closed. Polyiamonds are finite unions of these closed tiles, so they are compact. By Alexander duality [1]}, the number of holes of a polyiamond (the rank of the first homology group) is the number of finite bounded, connected components of the polyiamond's complement in the plane. The area of a hole is defined to be the number of tiles needed to fill it in.
